% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/nScree.R
\name{nScree}
\alias{nScree}
\title{Non Graphical Cattel's Scree Test}
\usage{
nScree(eig = NULL, x = eig, aparallel = NULL, cor = TRUE,
  model = "components", criteria = NULL, ...)
}
\arguments{
\item{eig}{depreciated parameter (use x instead): eigenvalues to analyse}

\item{x}{numeric: a \code{vector} of eigenvalues, a \code{matrix} of
correlations or of covariances or a \code{data.frame} of data}

\item{aparallel}{numeric: results of a parallel analysis.  Defaults
eigenvalues fixed at \eqn{\lambda >= \bar{\lambda}} (Kaiser and related
rule) or \eqn{\lambda >= 0} (CFA analysis)}

\item{cor}{logical: if \code{TRUE} computes eigenvalues from a correlation
matrix, else from a covariance matrix}

\item{model}{character: \code{"components"} or \code{"factors"}}

\item{criteria}{numeric: by default fixed at \eqn{\bar{\lambda}}.  When the
\eqn{\lambda}s are computed from a principal component analysis on a
correlation matrix, it corresponds to the usual Kaiser \eqn{\lambda >= 1}
rule. On a covariance matrix or from a factor analysis, it is simply the
mean.  To apply \eqn{\lambda >= 0}, sometimes used with factor analysis, fix
the criteria to \eqn{0}.}

\item{...}{variabe: additionnal parameters to give to the \code{cor} or
\code{cov} functions}
}
\value{
\item{Components }{ Data frame for the number of components/factors
according to different rules } \item{Components$noc }{ Number of
components/factors to retain according to optimal coordinates \emph{oc}}
\item{Components$naf }{ Number of components/factors to retain according to
the acceleration factor \emph{af}} \item{Components$npar.analysis }{Number
of components/factors to retain according to parallel analysis }
\item{Components$nkaiser }{ Number of components/factors to retain according
to the Kaiser rule } \item{Analysis }{ Data frame of vectors linked to the
different rules } \item{Analysis$Eigenvalues }{ Eigenvalues }
\item{Analysis$Prop }{ Proportion of variance accounted by eigenvalues }
\item{Analysis$Cumu }{ Cumulative proportion of variance accounted by
eigenvalues } \item{Analysis$Par.Analysis }{ Centiles of the random
eigenvalues generated by the parallel analysis. } \item{Analysis$Pred.eig }{
Predicted eigenvalues by each optimal coordinate regression line }
\item{Analysis$OC}{ Critical optimal coordinates \emph{oc}}
\item{Analysis$Acc.factor }{ Acceleration factor \emph{af}}
\item{Analysis$AF}{ Critical acceleration factor \emph{af}} Otherwise,
returns a summary of the analysis.
}
\description{
The \code{nScree} function returns an analysis of the number of component or
factors to retain in an exploratory principal component or factor analysis.
The function also returns information about the number of components/factors
to retain with the Kaiser rule and the parallel analysis.
}
\details{
The \code{nScree} function returns an analysis of the number of
components/factors to retain in an exploratory principal component or factor
analysis. Different solutions are given. The classical ones are the Kaiser
rule, the parallel analysis, and the usual scree test
(\code{\link{plotuScree}}).  Non graphical solutions to the Cattell
subjective scree test are also proposed: an acceleration factor (\emph{af})
and the optimal coordinates index \emph{oc}. The acceleration factor
indicates where the elbow of the scree plot appears. It corresponds to the
acceleration of the curve, i.e. the second derivative.  The optimal
coordinates are the extrapolated coordinates of the previous eigenvalue that
allow the observed eigenvalue to go beyond this extrapolation. The
extrapolation is made by a linear regression using the last eigenvalue
coordinates and the \eqn{k+1} eigenvalue coordinates. There are \eqn{k-2}
regression lines like this.  The Kaiser rule or a parallel analysis
criterion (\code{\link{parallel}}) must also be simultaneously satisfied to
retain the components/factors, whether for the acceleration factor, or for
the optimal coordinates.

If \eqn{\lambda_i} is the \eqn{i^{th}} eigenvalue, and \eqn{LS_i} is a
location statistics like the mean or a centile (generally the followings:
\eqn{1^{st}, \ 5^{th}, \ 95^{th}, \ or \ 99^{th}}).

The Kaiser rule is computed as: \deqn{ n_{Kaiser} = \sum_{i} (\lambda_{i}
\ge \bar{\lambda}).} Note that \eqn{\bar{\lambda}} is equal to 1 when a
correlation matrix is used.

The parallel analysis is computed as: \deqn{n_{parallel} = \sum_{i}
(\lambda_{i} \ge LS_i).}

The acceleration factor (\eqn{AF}) corresponds to a numerical solution to
the elbow of the scree plot: \deqn{n_{AF} \equiv \ If \ \left[ (\lambda_{i}
\ge LS_i) \ and \ max(AF_i) \right].}

The optimal coordinates (\eqn{OC}) corresponds to an extrapolation of the
preceeding eigenvalue by a regression line between the eigenvalue
coordinates and the last eigenvalue coordinates: \deqn{n_{OC} = \sum_i
\left[(\lambda_i \ge LS_i) \cap (\lambda_i \ge (\lambda_{i \ predicted})
\right].}
}
\examples{

## INITIALISATION
 data(dFactors)                      # Load the nFactors dataset
 attach(dFactors)
 vect         <- Raiche              # Uses the example from Raiche
 eigenvalues  <- vect$eigenvalues    # Extracts the observed eigenvalues
 nsubjects    <- vect$nsubjects      # Extracts the number of subjects
 variables    <- length(eigenvalues) # Computes the number of variables
 rep          <- 100                 # Number of replications for PA analysis
 cent         <- 0.95                # Centile value of PA analysis

## PARALLEL ANALYSIS (qevpea for the centile criterion, mevpea for the
## mean criterion)
 aparallel    <- parallel(var     = variables,
                          subject = nsubjects,
                          rep     = rep,
                          cent    = cent
                          )$eigen$qevpea  # The 95 centile

## NUMBER OF FACTORS RETAINED ACCORDING TO DIFFERENT RULES
 results      <- nScree(x=eigenvalues, aparallel=aparallel)
 results
 summary(results)

## PLOT ACCORDING TO THE nScree CLASS
 plotnScree(results)

}
\references{
Cattell, R. B. (1966). The scree test for the number of factors.
\emph{Multivariate Behavioral Research, 1}, 245-276.

Dinno, A. (2009). \emph{Gently clarifying the application of Horn's parallel
analysis to principal component analysis versus factor analysis}.  Portland,
Oregon: Portland Sate University.

Guttman, L. (1954). Some necessary conditions for common factor analysis.
\emph{Psychometrika, 19, 149-162}.

Horn, J. L. (1965). A rationale for the number of factors in factor
analysis.  \emph{Psychometrika, 30}, 179-185.

Kaiser, H. F. (1960). The application of electronic computer to factor
analysis.  \emph{Educational and Psychological Measurement, 20}, 141-151.

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions
for Cattell's scree test. Methodology, 9(1), 23-29.
}
\seealso{
\code{\link{plotuScree}}, \code{\link{plotnScree}},
\code{\link{parallel}}, \code{\link{plotParallel}},
}
\author{
Gilles Raiche \cr Centre sur les Applications des Modeles de
Reponses aux Items (CAMRI) \cr Universite du Quebec a Montreal\cr
\email{raiche.gilles@uqam.ca}
}
\keyword{multivariate}